Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

Odd Differences

Stage: 4 Challenge Level:

Tom started us off with the following
idea. Thanks Tom.

In the square of dots you can imagine the"L" of any odd number and
if you take the square for the odd number below it from the square
of the odd number you get the number you stated with. To find the
two squares you have to find half the odd number and round it up
and down. So half of $13$ is $6.5$ so the two squares have sides
$7$ and $6$.

Here is a picture:

So, using the diagram you know
that $$105= \frac{105-1}{2} + \frac{105+1}{2}$$

$$ 105 = 52 +53$$

$$ 105 = 53^2 - 52^2$$

You can always find one way using this method.
So

$$1155= \frac{1155-1}{2} +
\frac{1155+1}{2}$$

$$ 1155 = 577+578$$

$$ 105 = 578^2 - 577^2$$

Alex, from the Grammar School at Leeds extends
the algebra building on the second visualisation:

The task is to find eight pairs of numbers $(a,b)$, satisfying
the following equation: $$a^2 + b^2 =115$$

The LHS can be factorized to $(a+b)(a-b)$, so now we can begin
to find eight factor pairs of $1155$:

Using prime factorization,

$155 = 5 \times 231 = 5 \times7\times33 = 5 \times7 \times3
\times11$

Now we use different selections of these prime factors to
build the list of eight factor pairs:

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